An Application of Signal Detection Theory for Understanding Driver 

Behavior at Highway-Rail Grade Crossings 

Michelle Yeh and Jordan Multer 

United States Department of Transportation 

Volpe National Transportation Systems Center 

Cambridge, MA 

Thomas Raslear 

Federal Railroad Administration 

Washington, DC 

We used signal detection theory to examine if grade crossing warning devices were effective because 

they increased drivers’ sensitivity to a train’s approach or because they encouraged drivers to stop. We 

estimated d' and β for eight warning devices using 2006 data from the Federal Railroad 

Administration’s Highway-Rail Grade Crossing Accident/Incident database and Highway-Rail 

Crossing Inventory. We also calculated a measure of warning device effectiveness by comparing the 

maximum likelihood of an accident at a grade crossing with its observed probability. The 2006 results 

were compared to an earlier analysis of 1986 data. The collective findings indicate that grade crossing 

warning devices are effective because they encourage drivers to stop. Warning device effectiveness 

improved over the years, as drivers behaved more conservatively. Sensitivity also increased. The 

current model is descriptive, but it provides a framework for understanding driver decision-making at 

grade crossings and for examining the impact of proposed countermeasures. 

INTRODUCTION 

The Federal Railroad Administration (FRA) needs a 

better understanding of driver decision-making at highway-rail 

grade crossings. Grade crossing safety has improved; from 

1994 through 2003, the number of grade crossing accidents 

decreased by 41 percent and the number of fatalities fell by 48 

percent. However, many of the accidents that occurred were 

the result of driver error or poor judgment (Office of the 

Inspector General, 2004). Thus, we were interested in 

modeling drivers’ decision-making strategies to understand 

the factors that contribute to decisions at grade crossings and 

to establish a framework for research to evaluate the impact of 

proposed countermeasures. 

We applied signal detection theory (SDT) to describe 

drivers’ decisions whether to stop or proceed. In this 

application of SDT, the train is the signal, and it provides 

visual and auditory cues as to its approach (e.g., alerting lights 

and the sound of the horn). Other information at the grade 

crossing create noise that may compete with the signal, such 

as the flashing lights, gates, and bells at some crossings or 

sounds from inside the vehicle. 

There are two possible states of the world when a driver is 

approaching a grade crossing: that a train is approaching or 

that it is not. The driver must then make one of two choices: 

stop or proceed. A driver’s action at a grade crossing can thus 

be described by the 2 x 2 signal-response matrix in Table 1. 

Compliant behavior at grade crossings is indicated by the 

highlighted cells. A valid stop is the decision to stop when a 

train is approaching (a hit in SDT), and a correct crossing

describes the decision to proceed when a train is not 

approaching (a correct rejection). Of concern is a miss at a 

grade crossing—the decision to proceed when a train is close, 

which can result in an accident. Finally, a false stop, or false 

alarm, in the grade crossing situation results when a driver 

stops when no train is approaching. 

State of the World 

Train is close Train is not close 

Yes (Stop) 

Valid Stop 

(driver stops at 

crossing) 

False Stop 

(driver stops 

unnecessarily) 

Driver Response 

No 

(Proceed) 

Accident 

(driver does not 

stop) 

Correct Crossing 

(driver safely 

crosses tracks) 

Table 1. Signal-Response Matrix for a Driver at a Grade 

Crossing. 

Raslear (1996) applied this framework of driver decisionmaking to examine the effectiveness of eight grade crossing 

warning devices and to determine if the source of their 

effectiveness was because the warning devices increased the 

signal-to-noise ratio at the grade crossing (sensitivity) or 

because they encouraged drivers to stop (bias). The grade 

crossing systems consisted of four passive warning devices 

(no protection, the crossbuck, stop sign, or other signs or 

signals), classified as such because the information they 

provide does not change regardless of the presence or 

approach of a train, and four active warning devices (gates, 

flashing lights, highway traffic signals, and special warning 

devices). Device effectiveness was calculated for each 

warning device as a proportion of the accident risk at a grade 

crossing relative to the observed probability of an accident. 

The data used for the analysis were collected in 1986. 

The results indicated that active grade crossing warning 

devices were more effective than passive warning devices and 

that the source of this effectiveness was primarily the result of 

how they affected drivers’ bias. Active warning devices 

encouraged drivers to stop with gates having the highest bias. 
Grade crossings with no protection were the least effective, 

and drivers were more likely to proceed than to stop when 

encountering one of these crossings. The warning devices 

showed little difference in sensitivity, which was relatively 

high overall, suggesting that the train generally presents a 

tremendous signal relative to the background noise. 

More than 20 years later, safety at grade crossings has

improved significantly, and it was of interest to examine if 

these improvements could be captured using the SDT

framework. Thus, the purpose of this research was to update

the analysis conducted by Raslear and to compare our findings

to his to understand how driver decision-making at grade

crossings had changed. 

METHOD 

We estimated sensitivity, bias, and device effectiveness 

for eight grade crossing warning devices: 

• No protection,

• Other signs or signals, 

• Crossbuck,

• Stop sign, 

• Special warning devices, 

• Other activated warning devices (e.g., highway traffic 

signals, wigwags), 

• Flashing lights, and 

• Gates. 

The first four warning devices listed are considered passive

warning devices, and the last four are active warning devices. 

Sensitivity was estimated using d' and calculated as: 

d' = z[P(VS)] – z[P(FS)]. 

In the formula, P(VS) is the probability of a valid stop (a hit), 

and P(FS) is the probability of a false stop (a false alarm). Bias 

was estimated using β and calculated as: 

β = ( )

( )

y FS

y VS

, where 

[ ( )] / 2 2

( ) 0.3989 z P VS y VS e− = , and 

[ ( )] / 2 2

( ) 0.3989 z P FS y FS e− =

The values for P(VS) and P(FS) can be estimated from

accident data. An overview of the calculations is presented 

here. Additional details and formulas are provided in

Raslear (1996). 

By definition, P(VS) is equal to 1 minus the probability of

an accident (i.e., 1 – P(AC)). P(AC) can be calculated from the 

accident rate for each warning device, equalized for exposure. 

Each warning device is used at a different number of 

crossings, and each of these crossings is frequented by a 

different number of trains per day and a different number of

cars per day. Grade crossing warning devices are selected as a 

function of the number of trains and cars per day, so active 

warning devices will have higher exposure than passive

warning devices. To take all these factors into consideration 

when determining the probability of an accident, P(AC) was 

estimated for each grade crossing warning device as the 

accident rate per crossing per train per highway vehicle per 

minute. 

P(FS) was not as easy to define as P(VS). Rather, it was 

more straightforward to estimate the probability of a correct 

crossing, P(CC), and to calculate P(FS) as 1 minus P(CC).

P(CC) reflects the probability that a car and a train will not 

have an accident if a car does not stop. P(CC) can be estimated 

as 1 minus the maximum accident risk at a grade crossing, 

P(AC)max, where P(AC)max is the probability that a train and a 

car are at a crossing simultaneously and that neither can stop. 

Therefore: 

P(FS) = 1 – P(CC) = 1 – [1 - P(AC)max] = P(AC)max

P(FS) is equal to the accident risk associated with each 

warning device. 

P(AC)max was calculated as the product of the probability 

that one or more trains would be observed at a grade crossing 

in a 1-minute period (pT) and the probability that one or more 

highway vehicles would be observed at a grade crossing in a 

1-minute period (pH). The values for pT and pH can be 

estimated from the train rate per day at a crossing and the 

average annual daily traffic (AADT) at a crossing, 

respectively. Note that pT and pH are also used to estimate 

exposure for P(AC).

Finally, we also wanted to obtain a measure of device 

effectiveness. Simply examining the accident rate can not 

provide such a measure because the accident rate speaks only

to the observed frequency of accidents and not to the accident 

risk, which is the number of accidents that would have

occurred if no warning device was present. Thus, to estimate 

device effectiveness, Raslear (1996) computed the ratio 

between the maximum probability of an accident (the accident 

risk) and the observed probability of an accident (the accident 

rate). That is: 

Device Effectiveness = ( )

( )

P AC

P FS

The information needed for this analysis was contained in 

two separate FRA databases: the Highway-Rail Grade 

Crossing Accident/Incident database and the Highway-Rail 

Crossing Inventory. The Highway-Rail Grade Crossing

Accident/Incident database records details on each accident 

occurring at a highway-rail grade crossing per calendar year, 

and it was used to calculate the total number of accidents for 

each of the eight grade crossing warning devices. The 

Highway-Rail Crossing Inventory contains a list of all grade

crossings, and it was used to determine the number of public

grade crossings protected by each of the eight warning 

devices, the median number of trains per day for the crossings, 

and the median AADT at the crossings. 

Data from the Highway-Rail Grade Crossing

Accident/Incident database can be downloaded from the FRA 

Office of Safety web site for a specific year, but the HighwayRail Crossing Inventory does not provide a similar yearly

snapshot. Information from the databases is summarized in

FRA’s Annual Reports (e.g., see FRA (2008)), so data for the 

most recent year available—2006—were used for this

analysis. Data from 1986, the year for which Raslear (1996) 

conducted his analysis, were also examined as part of this 

effort for comparison. 

One problem with the data in the Highway-Rail Crossing

Inventory is that the fields for trains per day and AADT are
not updated each year. In fact, a comparison of the median 

AADT values recorded in 1986 to that recorded in 2006 

showed that AADT decreased for some warning devices. This 

finding was contrary to other data that showed an overall 

increase in the number of highway vehicles and national 

vehicle miles travelled (VMT), which would suggest a 

corresponding increase in AADT at grade crossings (see the 

Federal Highway Administration’s (FHWA) Highway 

Statistics series). Consequently, we were concerned with the 

reliability of the measures we used to equalize for exposure, 

and we sought ways to adjust our estimates for the train rate 

and highway vehicle rate per day at grade crossings. 

We used information regarding the increase in the overall 

number of trains from 1986 to 2006 to capture changes in the 

frequency of trains at a grade crossing. The number of trains 

was calculated by dividing the number of train miles traveled 

with the number of track miles for each year from 1986 

through 2006, using data from the FRA and the American 

Association of Railroads. The 1986 rate for trains per day 

from the Highway-Rail Crossing Inventory was not changed. 

Instead, the train rate per day for each warning device in 2006 

was estimated from the 1986 data and increased proportionally 

to reflect the change in the number of trains since 1986. We 

expected that this calculation would reflect the changes in 

exposure while minimizing the effect of any error in the train 

rate per day reported in the Highway-Rail Grade Crossing 

Inventory. 

A similar process was conducted to estimate in the 

highway vehicle rate. We collected information on national 

VMT from 1986 through 2006, as reported by FHWA. The 

AADT for 1986 was unchanged from that reported in the 

Highway-Rail Crossing Inventory, but the AADT values used 

for 2006 were calculated using the 1986 data as a baseline and 

increased proportionally to reflect the changes in VMT since 

1986. 

RESULTS 

In 1986, 5,710 accidents occurred at 192,454 grade 

crossings (approximately 0.030 accidents/crossing). The 

number of accidents dropped by over 50 percent by 2006, with 

2,387 accidents at 139,886 grade crossings (approximately 

0.017 accidents per crossing). During the same time period, 

exposure increased; the total trains per day at a grade crossing 

increased by 125 percent, and the AADT by 64 percent. This 

data was used to develop estimations for d', β, and device 

effectiveness for 1986 and 2006, as shown in Table 2. (Note 

that a direct comparison of these estimations to the overall 

change in the accident rate in the same time period is difficult 

because the estimations reflect probabilities.) Figure 1 shows a 

transformed Receiver-Operator Characteristic (ROC) curve for 

the data sets using a plot of z(VS) versus z(FS). Data for 1986 

are drawn in gray in the figure and data for 2006 in black. 

In first examining sensitivity, estimations of d' were fairly 

high for both years (6.95 in 1986 and 7.21 in 2006), 

suggesting that the train presents a fairly salient signal at a 

grade crossing. Proportionally, the change in sensitivity 

throughout the 20-year period was approximately 4 percent for 

all the warning devices combined with the greatest individual 

increase observed for the stop sign. This change, although 

small, was significant, t(7) = -3.16, p < 0.05. The mean d' for 

each year are represented by the dotted lines in Figure 1; note 

that because the figure shows z(VS) versus z(FS) rather than 

P(VS) versus P(FS), the sensitivity contours have a slope of 1. 

As the figure shows, the data points for each warning device 

for a given year fall fairly close to the mean for that year, thus 

indicating that little difference was in the signal-to-noise ratio 

across the warning devices. In 2006, sensitivity ranged from a 

low of 6.92 (gates) to a high of 7.54 (no signs or signals). The 

pattern of results makes sense intuitively; grade crossings with 

no signs or warning signals generate less perceptual “noise” 

that compete for the driver’s attention, so the signal-to-noise 

ratio may be greater and the train more salient. 

1986 2006 

Warning device d' β Device 

Effectiveness d' β Device Effectiveness

No Signs or Signals 7.40 1.49 0.66 7.54 0.07 16.41 

Other Signs or Signals 6.98 1.03 0.97 7.35 0.02 61.09 

Crossbuck 7.12 0.67 1.53 7.20 0.04 32.99 

Stop Signs 6.34 8.24 0.10 7.03 0.07 16.26 

Special Active Warning Devices 7.28 0.02 58.10 7.34 0.002 678.34 

Highway Traffic Signals, 

Wigwags, Bells, or Other 

Activated Warning Device 

6.68 0.15 7.75 7.19 0.002 1065.13 

Flashing Lights 6.96 0.01 141.18 7.14 0.001 3229.16 

Gates 6.84 0.001 1667.92 6.92 6.49E-05 31950.29 

AVERAGE 6.95 1.45 234.77 7.21 0.03 4629.11 

Table 2. Revised d', beta, and device effectiveness values for 1986 and 2006. 
0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Z(FS)

Z(VS)

Bias to stop

Bias to proceed

No bias

d' = 0

1986 2006

Gates

Flashing lights

Highw ay Traffic Signals

Special Active Warning Devices

No Protection

Crossbuck

Stop Sign

Other d' = 7.21

d' = 6.95

Figure 1. ROC plot: Adjusted values for 1986 and 2007. 

Although grade crossings with no warning devices have a 

high signal-to-noise ratio, the lack of information about the 

presence of a crossing and whether or not a train is 

approaching tends to encourage drivers to proceed through the 

crossing, as shown by the estimations for β. A value of β equal

to 1 represents no bias (in Figure 1, this is depicted as the solid

diagonal extending from the top left to point (0, 0)), a value of 

β less than 1 indicates a willingness to stop (the area above the 

diagonal), and a value of β greater than 1 indicates an 

inclination to proceed (the area below the diagonal). The 

average β in 1986 was 1.45, indicating that drivers were more

likely to proceed than to stop, but much of this risky behavior

reflected decisions at passive grade crossings. By 2006, 

drivers were more likely to stop than proceed at a grade

crossing; the estimations of β had improved from 1986 values 

by almost 50 times to an average of 0.03. A comparison of β

across the two years, using the natural logarithm, indicated

that this shift was significant, t(7) = 10.62, p < 0.05. This

conservative change in driver decision-making was observed 

for all the warning devices (in Figure 1, the greater willingness 

to stop can be seen in a shift up and to the right for each of the 

data points from 1986, and all the data points fall above the

solid diagonal where β = 1). In both years examined in this

analysis, the most conservative decisions were made at grade 

crossings protected with gates. This finding is not surprising, 

because gates are the most salient active protection system. In 

2006, the riskiest behavior occurred at grade crossings with no

signs or signals and those that were protected with stop signs, 

but even at these grade crossings, drivers exhibited a

willingness to stop (β = 0.07).

The changes in d' and β can be considered with respect to 

the device effectiveness ratios shown in Table 2. The higher

the ratio is above 1, the greater the effectiveness of the 

warning device. A ratio less than or equal to 1 indicates that a 

warning device is not effective because the accident rate is 

higher than the accident risk. As the data in Table 2 indicate, 

device effectiveness improved considerably from 1986 to

2006. As the accident risk increased, there was no 

corresponding increase in the accident rate. Active warning 

devices (the last four warning devices listed in the table) were 

in general more effective than passive warning devices (the

first four warning devices in the table). Gates were the most

effective warning device, with a device effectiveness ratio 10 

times greater than the second most effective warning device, 

flashing lights. For both years of the analysis, the stop sign

was the least effective warning device, although its 

effectiveness had greatly improved in 2006 (16.26) relative to

1986 (0.10), when there appeared to be a greater likelihood of

accidents than would be expected given the train and highway 

vehicle rates. 

We were interested in determining the source of the 

effectiveness of the warning devices to determine if changes in

the signal-to-noise ratio or changes in driver’s bias accounted 

for the increased device effectiveness. Correlations were 

performed between d' and device effectiveness and between β

and device effectiveness using the 2006 data. For the purposes 

of this analysis, device effectiveness was transformed using 

the logarithmic scale and β was transformed using the natural

logarithm. We found a statistically significant relationship 

between device effectiveness and β (r = -0.99, p < 0.05), 

suggesting that the setting of the response criterion accounted 

for most of the variance in device effectiveness. Changing the 

signal-to-noise ratio had little effect on device effectiveness; 

the correlation between device effectiveness and d' was not 

significant (r = -0.55, p > 0.05). The results are consistent with

those reported by Raslear (1996), who concluded that the

source of the effectiveness of grade crossing warning devices 

appears to be in how they influence the setting of bias in the 

decision-making process. 

DISCUSSION 

The results of the SDT analysis provide a glimpse into

how different warning devices influence drivers’ decisions at
grade crossings. The analysis showed that in the 20-year 

period examined, drivers became more conservative in their 

decisions at grade crossings, and this inclination to stop at 

grade crossings played a large role in improving the 

effectiveness of all warning devices by an order of magnitude. 

Sensitivity of the warning devices at grade crossings was high 

in 1986, and it increased only slightly but significantly 

through 2006. Thus, this change, although small, also 

contributed to reducing the accident risk at grade crossings, 

even though it was not reflected in device effectiveness. This 

finding makes sense intuitively; the warning devices are 

located at the grade crossing and are not part of the train, so 

they can not enhance the signal by themselves. More 

importantly, the results of the analysis also suggest that the 

warning devices did not contribute to increasing the noise at 

grade crossings. 

The estimations of d' and β also provide some insight into 

driver behavior at active versus passive grade crossings. The 

values of d' estimated for 2006 were only slightly higher at 

passive crossings relative to active grade crossings (7.28 vs. 

7.14), suggesting that the signal-to-noise ratio was fairly 

similar and the approach of a train was equally detectable. 

However, there was a much greater willingness to stop at 

active crossings (β = 0.001) than at passive crossings (β = 

0.05). This behavior contributed to a device effectiveness ratio 

almost 300 times greater for active warning devices than 

passive warning devices. Unfortunately, the model can not 

speak to the drivers’ motivations for stopping or proceeding. 

However, the pattern of results is consistent with the 

hypothesis that low train frequencies, which are more typical 

of passive crossings than active crossings, predispose drivers 

not to expect a train and thus biases them in general to 

proceed. 

Of the eight warning devices, the stop sign had the lowest 

estimate for d' and one of the highest for β. In other words, it 

was the least sensitive, but at these crossings, drivers made the 

most risky decisions. The use of a stop sign at grade crossings 

is controversial; the similarities between a driver’s decision at 

a grade crossing and that at a highway intersection initially 

prompted interest in its use, but the results of several 

observational studies have reported low rates of compliance 

(see Yeh and Multer, 2008). In fact, the estimate of β for 1986 

was 8.24, reflecting an inclination to proceed at stop signprotected crossings. Even though compliance improved by 

2006, the device effectiveness ratio was still almost equal to 

that of grade crossings with no warning device at all. 

Another concern with the use of the stop sign is that a 

driver, stopped at the crossing, may not be able to effectively 

judge the speed of an approaching train, because cues 

regarding the lateral movement of the train are not as 

available. The estimates for sensitivity may provide some 

evidence in this regard; if drivers were less able to detect cues 

of an approaching train when stopped at the crossing, we 

would expect to see a lower signal-to-noise ratio at these grade 

crossings than for those with warning devices that required 

drivers to slow on the approach (e.g., the crossbuck). Indeed, 

this is the case; estimates of d' for the stop sign in the two 

years examined were generally lower than the other passive 

warning devices. 

Thus, the current model can be used to provide insight 

into driver decision-making at grade crossings. Although 

descriptive in nature, the model can be used in conjunction 

with field studies or laboratory experiments to contribute to an 

understanding of driver behavior. Our next step is to apply the 

model to examine and quantify the impact of different 

countermeasures that have improved safety at grade crossings. 

Countermeasures can be classified into two categories: those 

that increase the detectability of the train (sensitivity), such as 

the use of alerting lights or reflecting train cars, and those that 

encourage drivers to stop (bias), such as imposing tangible 

penalties for grade crossing violations. We are particularly 

interested in trying to parse out the individual effects of the 

different countermeasures to examine the corresponding 

changes in sensitivity and bias. The results will be used to 

refine the SDT model, so it can be better applied to meet a 

variety of needs to understand and improve driver behavior. 

ACKNOWLEDGEMENTS 

This research was conducted with funding from the FRA’s 

Office of Research and Development. The views expressed 

herein are those of the authors and do not necessarily reflect 

the views of the John A. Volpe National Transportation 

Systems Center, the Research and Innovative Technology 

Administration, or the United States Department of 

Transportation. 

REFERENCES 

Federal Highway Administration. (2007). Highway Statistics 

2006. Washington, D.C.: Federal Highway 

Administration. Online at: 

http://www.fhwa.dot.gov/policy/ohpi/hss/index.cfm 

Federal Railroad Administration. (2008). Railroad Safety 

Statistics: 2006 Annual Report. Washington, D.C.: 

Federal Railroad Administration. Online at: 

http://safetydata.fra.dot.gov/officeofsafety/

Office of the Inspector General (2004). 2004 Audit of the 

Highway-Rail Grade Crossing Safety Program (Report 

Number: MH-2004-065). Washington, D.C.: Federal 

Railroad Administration, Office of the Inspector General. 

Raslear, T.G. (1996). Driver behavior at rail-highway grade 

crossings: A signal detection theory analysis. In A. A. 

Carroll and J. L. Helser (Eds.), Safety of highway-railroad 

grade crossings. Research needs workshop. Volume II – 

Appendices (Report No. DOT/FRA/ORD-95/14.2, DOTVNTSC-FRA-95-12.2, pp. F9-F56). Washington, DC: 

U.S. Department of Transportation. 

Yeh, M. and Multer, J. (2008). Driver Behavior at HighwayRailroad Grade Crossings: A Literature Review from 

1990–2006 (Report Nos. DOT-VNTSC-FRA-08-03 & 

DOT/FRA/ORD-08/03). Cambridge, MA: U.S. DOT 

Volpe Center.